Decoupling between SARS-CoV-2 transmissibility and population mobility associated with increasing immunity from vaccination and infection in South America

All South American countries from the Southern cone (Argentina, Brazil, Chile, Paraguay and Uruguay) experienced severe COVID-19 epidemic waves during early 2021 driven by the expansion of variants Gamma and Lambda, however, there was an improvement in different epidemic indicators since June 2021. To investigate the impact of national vaccination programs and natural infection on viral transmission in those South American countries, we analyzed the coupling between population mobility and the viral effective reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_t$$\end{document}Rt. Our analyses reveal that population mobility was highly correlated with viral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_t$$\end{document}Rt from January to May 2021 in all countries analyzed; but a clear decoupling occurred since May–June 2021, when the rate of viral spread started to be lower than expected from the levels of social interactions. These findings support that populations from the South American Southern cone probably achieved the conditional herd immunity threshold to contain the spread of regional SARS-CoV-2 variants circulating at that time.


A.1 Additional countries
In figures A.1 and A.2 we provide the same analysis shown in figures 1 and 2 respectively, for the case of Israel and Italy. It can be observed that the same phenomenon described in the main text takes place in these countries. Namely, a first period where the mobility successfully fits and predicts the reproductive number R t , and after the effect of the vaccination starts to complement the immunity from natural infection, a period where the R t decouples from the mobility. Figure A.1: Viral effective reproduction number R t and its estimationR t using mobility information. Background colors indicate the following time periods: in blue, the time period used to fit the linear model (see Section 4.2), in yellow, the period after the fitting, but before the decoupling point, and in red after the decoupling point. The black dot corresponds to the last time the reproductive number was above one. The correlation corresponds to the period used to fit the model. The delay indicated is the time-shift between the mobility time series and R t in order to maximize the correlation in the linear regression.

A.2.1 Estimating the reproduction rate
For the analysis in this paper we considered several possible estimations of the time-varying reproduction rate R t of the epidemic. We now briefly review the available approaches and explain our decision to settle on a given estimator for R t .
A first approach, proposed in [3], consists on estimating R t directly from incidence data I t , the number of observed new cases. The estimation proceeds as follows: assume that when a person becomes infected, it can start spawning new infections from its contacts. These new infections will be reflected in the number of cases (become detected) after a random period of time, which can be modeled by a probability distribution w s on s > 0. This is called the serial interval of the disease, and for SARS-CoV-2 it has been estimated by [4] as having a mean of 3.95 days and a standard deviation of 4.75 days. For definiteness we assume a discrete Gamma distribution for {w s } with these mean and variance.
Assuming that the number of contagions generated by an individual is Poisson and independent across individuals, the number of new infections at time t follows also a Poisson distribution given by: where R t is the current reproduction rate that we wish to estimate, and Λ t is given by: The intuition behind Λ t is that this should be the average number of new infections reported at time t for a reproduction rate of 1. The authors of [3] then propose to use a Bayesian approach. Assuming R t is approximately constant over a window of length τ , and that a priori is distributed as a Gamma random variable with shape parameter a and scale parameter b, the a posteriori distribution of R t can be computed and a suitable estimation of R t is obtained as: Eq.
(3) has a simple intuitive explanation: besides a small bias from the a priori parameters, it is the ratio between the new observed cases in a given time window to the number of expected cases for a reproduction rate of 1.
The main advantage of this method is that it makes little assumptions on the dynamics of the epidemic, only dealing with disease specific parameters and the reasonable Poisson assumption on contacts. The main disadvantage is that, in order to be robust against the noise in observations and cope with weekly seasonal effects observed in the data, we have to employ a pretty large estimating window τ (typically between 7 and 14 days). This introduces a significant lag in the estimation. With data up to time t, we are estimating the value of R t with a delay of up to 1 week. Since we are interested in the time correlations between mobility and the current value of the reproduction rate, this lag precludes us from using this robust estimator.
A second approach is proposed in [5], and is currently computed in real time for a list of countries in [1]. The authors assume a simple SIR model for the dynamics of currently active cases, which we denote by A t to avoid using the standard name I since we reserve I for incidence or new infections.
The dynamics of the active cases follow the evolution equation: Here 1/γ is the recovery rate, i.e. the average time a person stops spreading the disease. In the typical SIR model, R t = β t S t (N ), where β t is the current level of social interaction and S t /N the ratio of susceptible population. However, since we are interested in quantifying only the reproduction rate, we can employ directly eq. (4). A simple transformation of (4) expresses the growth rate of the active cases as a function of R t : Moreover, for small relative increments this can be further simplified using the approximation log(1 + x) ≈ x to write: where ∇ is the usual difference operator. Since most of the data available is for incidence of new cases I t , in order to construct the time series A t one resorts again to the SIR model equations to write: The complete procedure is as follows: given a time series data from case counts I t , construct the series of active cases using (7). Then model the growth rate evolution of A t by using a simple local level model [6] for the trend in the reproduction rate R t given by: where ε t , η t are measurement noises assumed Gaussian. A suitable estimation of R t can be directly obtained from eqs. (8) by applying the Kalman filtering technique [6]. The main advantages of this approach are two-fold: first they are extremely robust to noise in the measurements I t and consequently A t . The second advantage is that the estimate is real-time, i.e. it introduces no lag on the trend. This is perfectly suited for our purposes where we want to characterize the time correlation with the mobility estimation. During preparation of this manuscript, we enhanced the model in eqs. (8) to include cyclic components in order to model systematic weekly trends in the data. However, inclusion of these trends did not change significantly the estimated trend of the R t trajectory, so for all our analysis, we settled on the local level model approach and Kalman filtering of (8).

A.2.2 Backtesting/Validation of regression
Although the regression of the mobility features to the R t is fitted in the period with blue background in Figure 1, and therefore period with yellow background serves as a validation of this regression up to the decoupling time, in this section we provide further details and validation.
For all five countries, we took the period prior to the decoupling time, and divided it into two sections of approximately the same length. The first period is used to fit the regression parameters, and the second period is used to validate the fitting. Additionally, we compute the variance of the regression parameters in the standard way, and use them to draw Monte Carlo trajectories in order to build error bands around the regression curves. The results are shown in Figure A.3, where the correlation in both the training and validation period are also reported. The results tend to reasonably validate the obtained the regression.

A.2.3 Sensitivity to the threshold
As described in the main text, the decoupling time T D was defined as the moment when the coupling ratiô R t /R t definitely exceeds the threshold 1.10. In order to study the sensitivity to the value of this threshold, as well as to validate the methodology, we expose two additional experiments.
First, we present the decoupling time T D and the corresponding immunity cHIT for several values of the threshold. It can be observed in Table 1 that the obtained cHIT values fall inside the confidence intervals in Table 1, and actually they are are very close to the central cHIT values presented.
Second, we present a decoupling time detection by means of an alternative method. Figure A.